Thouraya Baranger

Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data

This paper is concerned with solving the Cauchy problem for an elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to adapted stopping criteria for the minimization process depending on the noise rate. Numerical examples involving smooth and singular data are presented.

Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE

Different numerical methods have been proposed in the literature for solving the Cauchy problem for linear elliptic equations modelling different physical phenomena (Laplace equation for the stationary heat equation, Lame operator in elasticity, etc.). The aim of this paper is to situate these methods, fixed point methods and domain decomposition based techniques in a general variational framework, and show the equivalence between them. A generalization of the energy gap method proposed by the authors for Fourier-like boundary conditions is studied. Then a comparison of these methods by using an analytical example and two numerical problems with complicated geometry or boundary conditions is performed in order to estimate their numerical performances. It appeared that no method stood out from the others due to better or worse performance. According to the situations, the classification of the performances in terms of conditioning, an essential factor because the Cauchy problem is ill-posed, or in terms of the capacity to deal with strongly spatially variable data, depends on the problem dealt with. The issue of regularization is not addressed here because it is method-dependent and distorts the appreciation of the basic performance of the different approaches.

Energy methods for Cauchy problems of evolutions equations

In the present paper a numerical method based on minimizing energy functionals, is developed for solving Cauchy problem of evolution equations. Two cases are treated: the parabolic equations as the heat equations and the hyperbolic ones as the elastodynamics equations. The method is first presented in some details, then, illustrated on various applications.

Data completion for linear symmetric operators as a Cauchy problem: An efficient method via energy-like error minimization

Data completion is a problem in which known or measured superabundant data exist for part of the boundaries of a domain, whereas the data for the rest of the boundaries are unknown. Thus the aim is to determine the solution of a known PDE defined throughout the domain, which satisfies the superabundant data and then identifies the missing ones. For linear symmetric operators, we propose a general method to solve the data completion problem as a Cauchy problem. Various applications are described for stationary conduction and elastostatic problems.

Efficient Model Development for an Assembled Rotor of an Induction Motor Using a Condensed Modal Functional

In order to predict the lateral rotordynamics of a high-speed induction motor, an optimization procedure is proposed for identifying the equivalent constitutive properties especially those of the magnetic core: an assembly of lamination stack, tie rods, and short-circuit rods. Modal parameters predicted by a finite element (FE) branched model based mainly on beam and disk elements and measured on an induction motor are included in an original energy functional. The minimization of this functional by using the Levenberg-Marquardt algorithm permits extracting the equivalent constitutive properties of the lamination stack.

Numerical analysis of an energy-like minimization method to solve a parabolic Cauchy problem with noisy data

This paper is concerned with solving the Cauchy problem for the parabolic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to an adapted stopping criteria depending on noise rate for the minimization process. Numerical experiments are performed and confirm the theoretical convergence order and the good behavior of the minimization process.

A criterion for mode shape tracking: application to Campbell diagrams

The correlation criterion proposed in this article and called nc2o (normalized cross complex orthogonality) is based on the bi-orthogonality properties between rotor mode shapes calculated at different speeds of rotation. This criterion is proved using an industrial laminated rotor composed of disks and two fluid film bearings, whose characteristics depend on the speed of rotation. The industrial finite element model shows that the nc2o criterion provides a more efficient mode pairing of rotor shapes than those obtained by using classical correlation criteria. Moreover this criterion makes it easier to plot Campbell diagrams for strongly speed-of-rotation-dependent structures.

Combined energy method and regularization to solve the Cauchy problem for the heat equation

This paper deals with an energy method coupled with total variation regularization and an adequate stopping criterion in order to solve a Cauchy problem for the heat equation when using noisy data. First, the Cauchy problem is written as a data completion one, then it is split into two well-posed thermal problems. Therefore, a pseudo-energy functional measuring the gap between solutions of these two problems is introduced and minimized. The problem is then converted into one of constrained optimization; the computation of the gradient of this functional is given for the full time–space discrete problems by means of the adjoint method. In order to deal with noisy data, two regularization techniques were used, the first one which fits into the optimization procedure is an adequate stopping criterion depending on the noise level and it avoids numerical instability. The second one is a total variation regularization method which can be carried out a priori and/or a posteriori of the optimization procedure. Numerical experiments are performed on the noisy Cauchy data and/or the identified data. Numerical experiments highlight the efficiency and weakness of the coupled methods.

On the Alternating Method for Cauchy Problems and Its Finite Element Discretisation

We consider the alternating method (Kozlov, V. A. and Maz’ya, V. G., On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz 1 (1989), 144–170. (English transl.: Leningrad Math. J. 1 (1990), 1207–1228.)) for the stable reconstruction of the solution to the Cauchy problem for the stationary heat equation in a bounded Lipschitz domain. Using results from Baranger, T. N. and Andrieux, S., (Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE, Appl. Math. Comput. 218 (2011), 1970–1989.), we show that the alternating method can be equivalently formulated as the minimisation of a certain gap functional, and we prove some properties of this functional and its minimum. It is shown that the original alternating method can be interpreted as a method for the solution of the Euler–Lagrange first-order optimality equations for the gap functional. Moreover, we show how to discretise this functional and equations via the finite element method (FEM). The error between the minimum of the continuous functional and the discretised one is investigated, and an estimate is given between these minima in terms of the mesh size and the error level in the data. Numerical examples are included showing that accurate reconstructions can be obtained also with a non-constant heat conductivity.

Identification of injection and extraction wells from overspecified boundary data

In this paper, we focus on the identification of wells’ positions and fluxes/flows from the knowledge of overspecified data: hydraulic head and flux, on a part of the domain boundary. The used method is based on minimizing a constitutive law gap functional. We consider two inverse problems: in the first one overspecified conditions are available throughout the entire domain boundary; in the second inverse problem, in addition to the wells, boundary condition are also unknown on an inaccessible part of the domain boundary.